Hair problems

If our anxious princess from last week loses 50 hairs a day, and the count stays at 150,000, that means she's growing 50 new ones every day. Logically, the ones that just emerged will be the last to fall out, so the average lifespan of a hair is 150,000:50 = 3,000 days. And since hair grows 1 cm per month, the princess's hair will measure, at its longest, about 1 meter.

But my astute readers couldn’t help but notice that the reality of hair is a bit more complex, so the riddle sparked numerous and very interesting comments, especially after a question posed by Salva Fuster: “Thinking again about the problem of Princess Amethyst, which I think is a very good one, the following question occurs to me: Could it be concluded that the length of hair remains approximately 1 meter if the growth phase of the hair lasted longer?” (And, by the way, a little meta-problem for insiders: How does SF know that the princess in the story is called Amethyst?)

On the other hand, when we talk about the length of a head of hair, do we mean the length of the longest hairs, the average length of all the hairs...? Or, as Rafael Granero says: “Paranoia leads me to see mice in statements like a dipsomaniac, I see cockroaches on the walls... Arithmetically, I conclude that if the head of hair has 150,000 hairs, distributed proportionally from those that are just born to those about to fall out, if the first ones measure almost nothing and the last ones almost a meter, then the head of hair measures 150,000 x 0.49 m”. That is, about 75 km.

How many hairs do you have?

Unless you're completely bald (or bald, which also happens), you probably don't know how many hairs you have on your head right now; but you might think it's a very personal endowment, and it's highly unlikely that anyone in your life has exactly the same number of hairs as you. And yet...

Considering that our princess's abundant hair represents a difficult-to-surpass maximum (normally around 100,000 hairs) and that Madrid has, roughly speaking, a population of 3.5 million, how many people in the capital, at least, have the same number of hairs on their heads? And at most?

A decade

I'm writing this installment of The Game of Science (no. 523) on June 12, 2025. And on another June 12, 2015, the first installment appeared, entitled The Buddhist Monk and the Fixed Point Theorem . Our section (not a majestic plural: I say this because it belongs to my kind readers/is as much or more than mine) is, therefore, ten years old, and it began by posing a question that seems more theological than logical-mathematical: "Could God, in the prelude to the Great Flood, have lashed the face of the Earth with an ubiquitous wind?" I'll leave it there again.

And since we've been focusing on trick questions in recent weeks, let's end with one related to our anniversary: ​​How many leap years can there be, at least and at most, in a decade?